A structure and asymptotic preserving scheme for the quasineutral limit of the Vlasov-Poisson system

TL;DR

The paper tackles numerical challenges in simulating the Vlasov–Poisson system in the quasineutral limit by introducing a Hermite spectral formulation in velocity and a structure-preserving, asymptotic-preserving (AP) discretization. It develops a fully discrete, two-step time-splitting scheme that remains uniformly accurate as the Debye length shrinks ($\\lambda \to 0$), and proves discrete convergence to the quasineutral limit, including a discrete reformulated Poisson equation. The analysis isolates slow and oscillatory components of the electric field, demonstrating that $E^{\\lambda}$ converges to $E_{ ext{slow}}$ up to fast $O(1/\\lambda)$ oscillations and that the oscillatory part can be rigorously controlled under suitable amplitude assumptions. Numerical experiments across near-equilibrium, smooth perturbations, oscillatory perturbations, and two-stream cases illustrate stability, asymptotic preservation, and the limitations of the quasineutral approximation for long-time dynamics, aligning with existing kinetic theory results. Overall, the approach delivers stable, efficient quasineutral simulations without requiring reformulation of Poisson into oscillator-type models, and confirms the method’s practical relevance for multiscale plasma dynamics.

Abstract

In this work, we propose a new numerical method for the Vlasov-Poisson system that is both asymptotically consistent and stable in the quasineutral regime, i.e. when the Debye length is small compared to the characteristic spatial scale of the physical domain. Our approach consists in reformulating the Vlasov-Poisson system as a hyperbolic problem by applying a spectral expansion in the basis of Hermite functions in the velocity space and in designing a structure-preserving scheme for the time and spatial variables. Through this Hermite formulation, we establish a convergence result for the electric field toward its quasineutral limit together with optimal error estimates. Following this path, we then propose a fully discrete numerical method for the Vlasov-Poisson system, inspired by the approach in arXiv:2306.14605 , and rigorously prove that it is uniformly consistent in the quasineutral limit regime. Finally, we present several numerical simulations to illustrate the behavior of the proposed scheme. These results demonstrate the capability of our method to describe quasineutral plasmas and confirm the theoretical findings: stability and asymptotic preservation.

Figures (13)

• Figure 4.1: Near equilibrium test case with $\alpha=0$ : (a) time evolution of the rescaled potential energy $\dfrac{1}{2}\sum_{j\in{\mathcal{J}}} \Delta x_j \,|E_j^n|^2$; (b) order of convergence for $\max_{n\geq 0}{\mathcal{E}}^\lambda_0(t^n)$ and $\max_{n\geq 0}{\mathcal{E}}^\lambda_1(t^n)$ ; (c) time evolution of ${\mathcal{E}}^\lambda_0(t)$ and $(d)$ time evolution of ${\mathcal{E}}^\lambda_1(t)$, as defined in \ref{['cE']}-\ref{['E:C_1:osc']}, for different values of $\lambda$.
• Figure 4.2: Near equilibrium test case with $\alpha=1/2$ : (a) time evolution of the rescaled potential energy $\dfrac{1}{2}\sum_{j\in{\mathcal{J}}} \Delta x_j \,|E_j^n|^2$; (b) order of convergence for $\max_{n\geq 0}{\mathcal{E}}^\lambda_0(t^n)$ and $\max_{n\geq 0}{\mathcal{E}}^\lambda_1(t^n)$ ; (c) time evolution of ${\mathcal{E}}^\lambda_0(t)$ and $(d)$ time evolution of ${\mathcal{E}}^\lambda_1(t)$, as defined in \ref{['cE']}-\ref{['E:C_1:osc']}, for different values of $\lambda$.
• Figure 4.3: Near equilibrium test case with $\alpha=1$ : (a) time evolution of the rescaled potential energy $\dfrac{1}{2}\sum_{j\in{\mathcal{J}}} \Delta x_j \,|E_j^n|^2$; (b) order of convergence for $\max_{n\geq 0}{\mathcal{E}}^\lambda_0(t^n)$ and $\max_{n\geq 0}{\mathcal{E}}^\lambda_1(t^n)$ ; (c) time evolution of ${\mathcal{E}}^\lambda_0(t)$ and $(d)$ time evolution of ${\mathcal{E}}^\lambda_1(t)$, as defined in \ref{['cE']}-\ref{['E:C_1:osc']}, for different values of $\lambda$.
• Figure 4.4: Near equilibrium test case. Time evolution of the rescaled potential energy $\dfrac{1}{2}\sum_{j\in{\mathcal{J}}} \Delta x_j \,|E_j^n|^2$ with $\Delta t=0.2$, $N_x=64$ and $N_H=128$ for different $\lambda$: (a) $\alpha=0$ ; (b) $\alpha=1/2$ ; (c) $\alpha=1$.
• Figure 4.5: Smooth perturbation of equilibrium test case: time evolution of $\|E^\lambda\|_{L^2}$ in logarithmic scale with $(a)$$\lambda= 1$; $(b)$$\lambda = 3. \;10^{-1}$; $(c)$$\lambda = 10^{-1}$ and $(d)$$\lambda = 3. 10^{-2}$.

Theorems & Definitions (7)

• Remark 1.1
• Proposition 2.1
• proof
• Remark 2.2
• Proposition 3.1
• Remark 3.2
• Proposition 3.3